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A256060
Queen Dido's puzzle (the founding of Carthage): a(n) is twice the maximal area of a polygon with 1) vertices on integral Cartesian coordinates, 2) no two edges parallel, and 3) all edge lengths less than or equal to n^2.
0
0, 0, 1, 1, 2, 36, 36, 36, 50, 53, 153, 153, 153, 333, 333, 333, 360
OFFSET
0,5
COMMENTS
The sequence may increase when n is the sum of two squares (A001481).
An optimal polygon will always be convex. - Gordon Hamilton
For parity reasons, the edges of the maximal-area polygon are not always as long as possible. This is true for a(9) through a(12). - Gordon Hamilton
This puzzle sequence could be used when introducing students to slopes.
Are these values known to be optimal or are they conjectures? - N. J. A. Sloane, Mar 13 2015
These values have not been proved to be optimal.
EXAMPLE
a(4) = 2 because this triangle has area 1 (remember a(n) is twice the area):
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a(5) = a(6) = a(7) = 36 because of this polygon of area 18:
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a(8) = 50 because of this polygon of area 25:
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a(9) = 53 because of this polygon of area 26.5:
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a(10) = 153 because of this polygon of area 76.5:
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CROSSREFS
Sequence in context: A095397 A353216 A073406 * A096513 A037418 A239343
KEYWORD
nonn,more
AUTHOR
Gordon Hamilton, Mar 13 2015
STATUS
approved